Lattice preferred orientation in omphacite: Examples from the Tauern Window, Austria and the Western Gneiss Region, Norway

It is lively debated how eclogites find their way from deep to mid-crustal levels during exhumation. Different exhumation models for high-pressure and ultrahigh-pressure rocks were suggested in previous studies, based mainly on field observations and less on microstructural studies on the exhumed rocks. The development and improvement of electron microscopy techniques allows it, to focus interest on direct investigations of microstructures and crystallographic properties in eclogites. In this case, it is of importance to study the applicability of crystallographic measurements on eclogites for exhumation processes and to unravel which processes affect eclogite textures. Previous studies suggested a strong relationship between deformation and lattice preferred orientation (LPO) in omphacite but it is still unclear if the deformation is related to the exhumation of eclogites. This study is focused on the questions which processes affect omphacite LPO and if textural investigations of omphacite are applicable for studying eclogite exhumation. Therefore, eclogites from two examples in the Alps and in the Caledonides were collected systematically and investigated with respect to omphacite LPO by using the electron backscattered diffraction (EBSD) technique. Omphacite textures of the Tauern Window (Austria) and the Western Gneiss Region (Norway) were studied to compare lattice preferred orientation with field observations and suggested exhumation models from previous studies. The interpretation of omphacite textures, regarding the deformation regime is mainly based on numerical simulations in previous studies. Omphacite LPO patterns of the Eclogite Zone are clearly independent from any kind of exhumation process. The textures were generated during omphacite growth on the prograde path of eclogite development until metamorphic peak conditions. Field observations in the Eclogite Zone show that kinematics in garnet mica schist, surrounding the eclogites, strongly indicate an extrusion wedge geometry. Stretching lineations show top-N thrusting at the base and a top-S normal faulting with a sinistral shear component at the top of the Eclogite Zone. The different shear sense on both sides of the unit does not affect the omphacite textures in any way. The omphacite lattice preferred orientation patterns of the Western Gneiss Region can not be connected with any exhumation model. The textures were probably generated during the metamorphic peak and reflect the change from subduction to exhumation. Eclogite Zone and Western Gneiss Region differ significantly in size and especially in metamorphic conditions. While the Eclogite Zone is characterized by constant P-T conditions (600-650°C, 20-25 kbar), the Western Gneiss Region contains a wide P-T range from high- to ultrahigh pressure conditions (400-800°C, 20-35 kbar). In contrast to this, the omphacite textures of both units are very similar. This means that omphacite LPO is independent from P-T conditions and therefore from burial depth. Further, in both units, omphacite LPO is independent from grain and subgrain size as well as from any shape preferred orientation (SPO) on grain and subgrain scale. Overall, omphacite lattice preferred orientation are generated on the prograde part of omphacite development. Therefore, textural investigations on omphacite LPO are not applicable to study eclogite exhumation.

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Fluid-structure interaction by a coupled lattice Boltzmann-finite element approach

In this thesis, a strategy to model the behavior of fluids and their interaction with deformable bodies is proposed. The fluid domain is modeled by using the lattice Boltzmann method, thus analyzing the fluid dynamics by a mesoscopic point of view. It has been proved that the solution provided by this method is equivalent to solve the Navier-Stokes equations for an incompressible flow with a second-order accuracy. Slender elastic structures idealized through beam finite elements are used. Large displacements are accounted for by using the corotational formulation. Structural dynamics is computed by using the Time Discontinuous Galerkin method. Therefore, two different solution procedures are used, one for the fluid domain and the other for the structural part, respectively. These two solvers need to communicate and to transfer each other several information, i.e. stresses, velocities, displacements. In order to guarantee a continuous, effective, and mutual exchange of information, a coupling strategy, consisting of three different algorithms, has been developed and numerically tested. In particular, the effectiveness of the three algorithms is shown in terms of interface energy artificially produced by the approximate fulfilling of compatibility and equilibrium conditions at the fluid-structure interface. The proposed coupled approach is used in order to solve different fluid-structure interaction problems, i.e. cantilever beams immersed in a viscous fluid, the impact of the hull of the ship on the marine free-surface, blood flow in a deformable vessels, and even flapping wings simulating the take-off of a butterfly. The good results achieved in each application highlight the effectiveness of the proposed methodology and of the C++ developed software to successfully approach several two-dimensional fluid-structure interaction problems.

Permutation classes, sorting algorithms, and lattice paths

A permutation is said to avoid a pattern if it does not contain any subsequence which is order-isomorphic to it. Donald Knuth, in the first volume of his celebrated book "The art of Computer Programming", observed that the permutations that can be computed (or, equivalently, sorted) by some particular data structures can be characterized in terms of pattern avoidance. In more recent years, the topic was reopened several times, while often in terms of sortable permutations rather than computable ones. The idea to sort permutations by using one of Knuth’s devices suggests to look for a deterministic procedure that decides, in linear time, if there exists a sequence of operations which is able to convert a given permutation into the identical one. In this thesis we show that, for the stack and the restricted deques, there exists an unique way to implement such a procedure. Moreover, we use these sorting procedures to create new sorting algorithms, and we prove some unexpected commutation properties between these procedures and the base step of bubblesort. We also show that the permutations that can be sorted by a combination of the base steps of bubblesort and its dual can be expressed, once again, in terms of pattern avoidance. In the final chapter we give an alternative proof of some enumerative results, in particular for the classes of permutations that can be sorted by the two restricted deques. It is well-known that the permutations that can be sorted through a restricted deque are counted by the Schrӧder numbers. In the thesis, we show how the deterministic sorting procedures yield a bijection between sortable permutations and Schrӧder paths.

Interacting bosons and fermions in three-dimensional optical lattice potentials

This thesis reports on the realization, characterization and analysis of ultracold bosonic and fermionic atoms in three-dimensional optical lattice potentials. Ultracold quantum gases in optical lattices can be regarded as ideal model systems to investigate quantum many-body physics. In this work interacting ensembles of bosonic 87Rb and fermionic 40K atoms are employed to study equilibrium phases and nonequilibrium dynamics. The investigations are enabled by a versatile experimental setup, whose core feature is a blue-detuned optical lattice that is combined with Feshbach resonances and a red-detuned dipole trap to allow for independent control of tunneling, interactions and external confinement. The Fermi-Hubbard model, which plays a central role in the theoretical description of strongly correlated electrons, is experimentally realized by loading interacting fermionic spin mixtures into the optical lattice. Using phase-contrast imaging the in-situ size of the atomic density distribution is measured, which allows to extract the global compressibility of the many-body state as a function of interaction and external confinement. Thereby, metallic and insulating phases are clearly identified. At strongly repulsive interaction, a vanishing compressibility and suppression of doubly occupied lattice sites signal the emergence of a fermionic Mott insulator. In a second series of experiments interaction effects in bosonic lattice quantum gases are analyzed. Typically, interactions between microscopic particles are described as two-body interactions. As such they are also contained in the single-band Bose-Hubbard model. However, our measurements demonstrate the presence of multi-body interactions that effectively emerge via virtual transitions of atoms to higher lattice bands. These findings are enabled by the development of a novel atom optical measurement technique: In quantum phase revival spectroscopy periodic collapse and revival dynamics of the bosonic matter wave field are induced. The frequencies of the dynamics are directly related to the on-site interaction energies of atomic Fock states and can be read out with high precision. The third part of this work deals with mixtures of bosons and fermions in optical lattices, in which the interspecies interactions are accurately controlled by means of a Feshbach resonance. Studies of the equilibrium phases show that the bosonic superfluid to Mott insulator transition is shifted towards lower lattice depths when bosons and fermions interact attractively. This observation is further analyzed by applying quantum phase revival spectroscopy to few-body systems consisting of a single fermion and a coherent bosonic field on individual lattice sites. In addition to the direct measurement of Bose-Fermi interaction energies, Bose-Bose interactions are proven to be modified by the presence of a fermion. This renormalization of bosonic interaction energies can explain the shift of the Mott insulator transition. The experiments of this thesis lay important foundations for future studies of quantum magnetism with fermionic spin mixtures as well as for the realization of complex quantum phases with Bose-Fermi mixtures. They furthermore point towards physics that reaches beyond the single-band Hubbard model.

Su(3) lattice gauge theories and spin chains

I modelli su reticolo con simmetrie SU(n) sono attualmente oggetto di studio sia dal punto di vista sperimentale, sia dal punto di vista teorico; particolare impulso alla ricerca in questo campo è stato dato dai recenti sviluppi in campo sperimentale per quanto riguarda la tecnica dell’intrappolamento di atomi ultrafreddi in un reticolo ottico. In questa tesi viene studiata, sia con tecniche analitiche sia con simulazioni numeriche, la generalizzazione del modello di Heisenberg su reticolo monodimensionale a simmetria SU(3). In particolare, viene proposto un mapping tra il modello di Heisenberg SU(3) e l’Hamiltoniana con simmetria SU(2) bilineare-biquadratica con spin 1. Vengono inoltre presentati nuovi risultati numerici ottenuti con l’algoritmo DMRG che confermano le previsioni teoriche in letteratura sul modello in esame. Infine è proposto un approccio per la formulazione della funzione di partizione dell’Hamiltoniana bilineare-biquadratica a spin-1 servendosi degli stati coerenti per SU(3).

Imaging and modeling of pore scale processes in porous media using X-ray computed tomography and lattice Boltzmann solver

In der Erdöl– und Gasindustrie sind bildgebende Verfahren und Simulationen auf der Porenskala im Begriff Routineanwendungen zu werden. Ihr weiteres Potential lässt sich im Umweltbereich anwenden, wie z.B. für den Transport und Verbleib von Schadstoffen im Untergrund, die Speicherung von Kohlendioxid und dem natürlichen Abbau von Schadstoffen in Böden. Mit der Röntgen-Computertomografie (XCT) steht ein zerstörungsfreies 3D bildgebendes Verfahren zur Verfügung, das auch häufig für die Untersuchung der internen Struktur geologischer Proben herangezogen wird. Das erste Ziel dieser Dissertation war die Implementierung einer Bildverarbeitungstechnik, die die Strahlenaufhärtung der Röntgen-Computertomografie beseitigt und den Segmentierungsprozess dessen Daten vereinfacht. Das zweite Ziel dieser Arbeit untersuchte die kombinierten Effekte von Porenraumcharakteristika, Porentortuosität, sowie die Strömungssimulation und Transportmodellierung in Porenräumen mit der Gitter-Boltzmann-Methode. In einer zylindrischen geologischen Probe war die Position jeder Phase auf Grundlage der Beobachtung durch das Vorhandensein der Strahlenaufhärtung in den rekonstruierten Bildern, das eine radiale Funktion vom Probenrand zum Zentrum darstellt, extrahierbar und die unterschiedlichen Phasen ließen sich automatisch segmentieren. Weiterhin wurden Strahlungsaufhärtungeffekte von beliebig geformten Objekten durch einen Oberflächenanpassungsalgorithmus korrigiert. Die Methode der „least square support vector machine” (LSSVM) ist durch einen modularen Aufbau charakterisiert und ist sehr gut für die Erkennung und Klassifizierung von Mustern geeignet. Aus diesem Grund wurde die Methode der LSSVM als pixelbasierte Klassifikationsmethode implementiert. Dieser Algorithmus ist in der Lage komplexe geologische Proben korrekt zu klassifizieren, benötigt für den Fall aber längere Rechenzeiten, so dass mehrdimensionale Trainingsdatensätze verwendet werden müssen. Die Dynamik von den unmischbaren Phasen Luft und Wasser wird durch eine Kombination von Porenmorphologie und Gitter Boltzmann Methode für Drainage und Imbibition Prozessen in 3D Datensätzen von Böden, die durch synchrotron-basierte XCT gewonnen wurden, untersucht. Obwohl die Porenmorphologie eine einfache Methode ist Kugeln in den verfügbaren Porenraum einzupassen, kann sie dennoch die komplexe kapillare Hysterese als eine Funktion der Wassersättigung erklären. Eine Hysterese ist für den Kapillardruck und die hydraulische Leitfähigkeit beobachtet worden, welche durch die hauptsächlich verbundenen Porennetzwerke und der verfügbaren Porenraumgrößenverteilung verursacht sind. Die hydraulische Konduktivität ist eine Funktion des Wassersättigungslevels und wird mit einer makroskopischen Berechnung empirischer Modelle verglichen. Die Daten stimmen vor allem für hohe Wassersättigungen gut überein. Um die Gegenwart von Krankheitserregern im Grundwasser und Abwässern vorhersagen zu können, wurde in einem Bodenaggregat der Einfluss von Korngröße, Porengeometrie und Fluidflussgeschwindigkeit z.B. mit dem Mikroorganismus Escherichia coli studiert. Die asymmetrischen und langschweifigen Durchbruchskurven, besonders bei höheren Wassersättigungen, wurden durch dispersiven Transport aufgrund des verbundenen Porennetzwerks und durch die Heterogenität des Strömungsfeldes verursacht. Es wurde beobachtet, dass die biokolloidale Verweilzeit eine Funktion des Druckgradienten als auch der Kolloidgröße ist. Unsere Modellierungsergebnisse stimmen sehr gut mit den bereits veröffentlichten Daten überein.

Helicity methods in LO and NLO QCD calculations

The goal of this thesis is the acceleration of numerical calculations of QCD observables, both at leading order and next–to–leading order in the coupling constant. In particular, the optimization of helicity and spin summation in the context of VEGAS Monte Carlo algorithms is investigated. In the literature, two such methods are mentioned but without detailed analyses. Only one of these methods can be used at next–to–leading order. This work presents a total of five different methods that replace the helicity sums with a Monte Carlo integration. This integration can be combined with the existing phase space integral, in the hope that this causes less overhead than the complete summation. For three of these methods, an extension to existing subtraction terms is developed which is required to enable next–to–leading order calculations. All methods are analyzed with respect to efficiency, accuracy, and ease of implementation before they are compared with each other. In this process, one method shows clear advantages in relation to all others.

Efficient algorithms for NLO QCD event generators

In this thesis we present techniques that can be used to speed up the calculation of perturbative matrix elements for observables with many legs ($n = 3, 4, 5, 6, 7, ldots$). We investigate several ways to achieve this, including the use of Monte Carlo methods, the leading-color approximation, numerically less precise but faster operations, and SSE-vectorization. An important idea is the use of enquote{random polarizations} for which we derive subtraction terms for the real corrections in next-to-leading order calculations. We present the effectiveness of all these methods in the context of electron-positron scattering to $n$ jets, $n$ ranging from two to seven.

Edge states and zero modes in quadratic fermionic models on a 1-D lattice

In una formulazione rigorosa della teoria quantistica, la definizione della varietà Riemanniana spaziale su cui il sistema è vincolato gioca un ruolo fondamentale. La presenza di un bordo sottolinea l'aspetto quantistico del sistema: l'imposizione di condizioni al contorno determina la discretizzazione degli autovalori del Laplaciano, come accade con condizioni note quali quelle periodiche, di Neumann o di Dirichlet. Tuttavia, non sono le uniche possibili. Qualsiasi condizione al bordo che garantisca l'autoaggiunzione dell' operatore Hamiltoniano è ammissibile. Tutte le possibili boundary conditions possono essere catalogate a partire dalla richiesta di conservazione del flusso al bordo della varietà. Alcune possibili condizioni al contorno, permettono l'esistenza di stati legati al bordo, cioè autostati dell' Hamiltoniana con autovalori negativi, detti edge states. Lo scopo di questa tesi è quello di investigare gli effetti di bordo in sistemi unidimensionali implementati su un reticolo discreto, nella prospettiva di capire come simulare proprietà di edge in un reticolo ottico. Il primo caso considerato è un sistema di elettroni liberi. La presenza di edge states è completamente determinata dai parametri di bordo del Laplaciano discreto. Al massimo due edge states emergono, e possono essere legati all' estremità destra o sinistra della catena a seconda delle condizioni al contorno. Anche il modo in cui decadono dal bordo al bulk e completamente determinato dalla scelta delle condizioni. Ammettendo un' interazione quadratica tra siti primi vicini, un secondo tipo di stati emerge in relazione sia alle condizioni al contorno che ai parametri del bulk. Questi stati sono chiamati zero modes, in quanto esiste la possibilità che siano degeneri con lo stato fondamentale. Per implementare le più generali condizioni al contorno, specialmente nel caso interagente, è necessario utilizzare un metodo generale per la diagonalizzazione, che estende la tecnica di Lieb-Shultz-Mattis per Hamiltoniane quadratiche a matrici complesse.

Quantum simulation of (1+1)D QED via a Zn lattice Gauge theory

La simulazione di un sistema quantistico complesso rappresenta ancora oggi una sfida estremamente impegnativa a causa degli elevati costi computazionali. La dimensione dello spazio di Hilbert cresce solitamente in modo esponenziale all'aumentare della taglia, rendendo di fatto impossibile una implementazione esatta anche sui più potenti calcolatori. Nel tentativo di superare queste difficoltà, sono stati sviluppati metodi stocastici classici, i quali tuttavia non garantiscono precisione per sistemi fermionici fortemente interagenti o teorie di campo in regimi di densità finita. Di qui, la necessità di un nuovo metodo di simulazione, ovvero la simulazione quantistica. L'idea di base è molto semplice: utilizzare un sistema completamente controllabile, chiamato simulatore quantistico, per analizzarne un altro meno accessibile. Seguendo tale idea, in questo lavoro di tesi si è utilizzata una teoria di gauge discreta con simmetria Zn per una simulazione dell'elettrodinamica quantistica in (1+1)D, studiando alcuni fenomeni di attivo interesse di ricerca, come il diagramma di fase o la dinamica di string-breaking, che generalmente non sono accessibili mediante simulazioni classiche. Si propone un diagramma di fase del modello caratterizzato dalla presenza di una fase confinata, in cui emergono eccitazioni mesoniche ed antimesoniche, cioè stati legati particella-antiparticella, ed una fase deconfinata.

The bulk transition of many-flavour QCD and the search for a UVFP at strong coupling

We explore the nature of the bulk transition observed at strong coupling in the SU(3) gauge theory with Nf=12 fermions in the fundamental representation. The transition separates a weak coupling chirally symmetric phase from a strong coupling chirally broken phase and is compatible with the scenario where conformality is restored by increasing the flavour content of a non abelian gauge theory. We explore the intriguing possibility that the observed bulk transition is associated with the occurrence of an ultraviolet fixed point (UVFP) at strong coupling, where a new theory emerges in the continuum.